Periodic travelling waves in convex Klein-Gordon chains. (English) Zbl 1175.37075
Summary: We study Klein-Gordon chains with attractive nearest neighbour forces and convex on-site potential, and show that there exists a two-parameter family of periodic travelling waves (wave trains) with unimodal and even profile functions. Our existence proof is based on a saddle-point problem with constraints and exploits the invariance properties of an improvement operator. Finally, we discuss the numerical computation of wave trains.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 47J30 | Variational methods involving nonlinear operators |
| 70F45 | The dynamics of infinite particle systems |
| 74J30 | Nonlinear waves in solid mechanics |
References:
| [1] | Dauxois, T.; Peyrard, M.; Bishop, A. R., Dynamics and thermodynamics of a nonlinear model for DNA denaturation, Phys. Rev. E, 47, 1, 684-695 (1993) |
| [2] | Braun, O. M.; Kivshar, Y. S., The Frenkel-Kontorova Model: Concepts, Methods, and Applications. Theoretical and Mathematical Physics (2004), Springer · Zbl 1140.82001 |
| [3] | Iooss, G.; James, G., Localized waves in nonlinear oscillator chains, Chaos, 15, 1 (2005), 015113: 1-15 · Zbl 1080.37080 |
| [4] | Iooss, G.; Kirchgässner, K., Travelling waves in a chain of coupled nonlinear oscillators, Comm. Math. Phys., 211, 2, 439-464 (2000) · Zbl 0956.37055 |
| [5] | Iooss, G.; Pelinovsky, D. E., Normal form for travelling kinks in discrete Klein Gordon lattices, Physica D, 616, 2, 327-345 (2006) · Zbl 1102.37045 |
| [6] | Duncan, D. B.; Eilbeck, J. C.; Feddersen, H.; Wattis, J. A.D., Solitons on lattices, Physica D, 68, 1-11 (1993) · Zbl 0785.35087 |
| [7] | Morgante, A. M.; Johansson, M.; Kopidakis, G.; Aubry, S., Standing wave instabilities in a chain of nonlinear coupled oscillators, Physica D, 162, 1/2, 53-94 (2002) · Zbl 1005.70020 |
| [8] | Katriel, G., Existence of travelling waves in discrete Sine-Gordon rings, SIAM J. Math. Anal., 36, 5, 1434-1443 (2005) · Zbl 1086.34027 |
| [9] | Bates, P. W.; Zhang, C., Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction, Discrete. Contin. Dyn. Syst. Ser. A, 16, 1, 235-252 (2006) · Zbl 1119.34034 |
| [10] | C.F. Kreiner, J. Zimmer, Travelling wave solutions for the discrete Sine-Gordon equation with nonlinear pair interaction, BICS preprint 3/08, 2008; C.F. Kreiner, J. Zimmer, Travelling wave solutions for the discrete Sine-Gordon equation with nonlinear pair interaction, BICS preprint 3/08, 2008 · Zbl 1160.37415 |
| [11] | C.F. Kreiner, J. Zimmer, Heteroclinic travelling waves for the lattice Sine-Gordon equation with linear pair interaction, BICS preprint 1/08, 2008; C.F. Kreiner, J. Zimmer, Heteroclinic travelling waves for the lattice Sine-Gordon equation with linear pair interaction, BICS preprint 1/08, 2008 · Zbl 1185.37156 |
| [12] | Pankov, A., Traveling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices (2005), Imperial College Press: Imperial College Press London · Zbl 1088.35001 |
| [13] | M. Herrmann, Unimodal wave trains and solitons in convex FPU chains, arXiv:0901.3736, 2008; M. Herrmann, Unimodal wave trains and solitons in convex FPU chains, arXiv:0901.3736, 2008 |
| [14] | M. Herrmann, J. Rademacher, Heteroclinic travelling waves in convex FPU-type chains, arXiv:0812.1712, 2008; M. Herrmann, J. Rademacher, Heteroclinic travelling waves in convex FPU-type chains, arXiv:0812.1712, 2008 · Zbl 1232.37040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
